Best Known (80−78, 80, s)-Nets in Base 32
(80−78, 80, 44)-Net over F32 — Constructive and digital
Digital (2, 80, 44)-net over F32, using
- t-expansion [i] based on digital (1, 80, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
(80−78, 80, 53)-Net over F32 — Digital
Digital (2, 80, 53)-net over F32, using
- net from sequence [i] based on digital (2, 52)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 2 and N(F) ≥ 53, using
(80−78, 80, 98)-Net over F32 — Upper bound on s (digital)
There is no digital (2, 80, 99)-net over F32, because
- 14 times m-reduction [i] would yield digital (2, 66, 99)-net over F32, but
- extracting embedded orthogonal array [i] would yield linear OA(3266, 99, F32, 64) (dual of [99, 33, 65]-code), but
- residual code [i] would yield OA(322, 34, S32, 2), but
- bound for OAs with strength k = 2 [i]
- the Rao or (dual) Hamming bound shows that M ≥ 1055 > 322 [i]
- residual code [i] would yield OA(322, 34, S32, 2), but
- extracting embedded orthogonal array [i] would yield linear OA(3266, 99, F32, 64) (dual of [99, 33, 65]-code), but
(80−78, 80, 176)-Net in Base 32 — Upper bound on s
There is no (2, 80, 177)-net in base 32, because
- extracting embedded orthogonal array [i] would yield OA(3280, 177, S32, 78), but
- the linear programming bound shows that M ≥ 642436 233588 222889 547956 035134 666783 015807 699005 218396 284416 626888 198721 507807 950436 092711 209909 054383 473409 289754 927761 778435 686400 / 245835 164771 > 3280 [i]